scitbx::matrix::tensor_rank_2 Namespace Reference

Computations involving tensors of rank 2. More...

Functions
template<typename NumTypeA, typename NumTypeG>
sym_mat3< NumTypeG >	gradient_transform (mat3< NumTypeA > const &a, sym_mat3< NumTypeG > const &g)
	Transformation of gradients (first derivatives) w.r.t. the elements of a symmetric tensor.
template<typename NumTypeA>
void	gradient_transform_matrix (NumTypeA *result, NumTypeA const *a)
	See overload.
template<typename NumTypeA>
af::versa< NumTypeA, af::c_grid< 2 > >	gradient_transform_matrix (mat3< NumTypeA > const &a)
	Transformation matrix for gradients (first derivatives) w.r.t. the elements of a symmetric tensor.

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Detailed Description

Computations involving tensors of rank 2.

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Function Documentation

sym_mat3<NumTypeG> scitbx::matrix::tensor_rank_2::gradient_transform	(	mat3< NumTypeA > const &	a,
		sym_mat3< NumTypeG > const &	g
	)			[inline]

Transformation of gradients (first derivatives) w.r.t. the elements of a symmetric tensor.

Parameters:

	a	is a linear transformation.
	g	are the gradients.

Mathematica script used to determine the transformation law:

        g={{g0,g3,g4},{g3,g1,g5},{g4,g5,g2}}
        a={{a0,a1,a2},{a3,a4,a5},{a6,a7,a8}}
        agat = a.g.Transpose[a]
        D[agat, g0]
        D[agat, g1]
        D[agat, g2]
        D[agat, g3]
        D[agat, g4]
        D[agat, g5]

Output:

            2                           2                           2
        {{a0 , a0 a3, a0 a6}, {a0 a3, a3 , a3 a6}, {a0 a6, a3 a6, a6 }}

            2                           2                           2
        {{a1 , a1 a4, a1 a7}, {a1 a4, a4 , a4 a7}, {a1 a7, a4 a7, a7 }}

            2                           2                           2
        {{a2 , a2 a5, a2 a8}, {a2 a5, a5 , a5 a8}, {a2 a8, a5 a8, a8 }}

        {{2 a0 a1, a1 a3 + a0 a4, a1 a6 + a0 a7},
         {a1 a3 + a0 a4, 2 a3 a4, a4 a6 + a3 a7},
         {a1 a6 + a0 a7, a4 a6 + a3 a7, 2 a6 a7}}

        {{2 a0 a2, a2 a3 + a0 a5, a2 a6 + a0 a8},
         {a2 a3 + a0 a5, 2 a3 a5, a5 a6 + a3 a8},
         {a2 a6 + a0 a8, a5 a6 + a3 a8, 2 a6 a8}}

        {{2 a1 a2, a2 a4 + a1 a5, a2 a7 + a1 a8},
         {a2 a4 + a1 a5, 2 a4 a5, a5 a7 + a4 a8},
         {a2 a7 + a1 a8, a5 a7 + a4 a8, 2 a7 a8}}
      

The transformed gradients are obtained by the dot products of the upper triangles of the derivative matrices with the original gradients g.

Referenced by cctbx::adptbx::grad_u_cart_as_u_star(), and cctbx::adptbx::grad_u_star_as_u_cart().

af::versa<NumTypeA, af::c_grid<2> > scitbx::matrix::tensor_rank_2::gradient_transform_matrix

(

mat3< NumTypeA > const &

a

)

[inline]

Transformation matrix for gradients (first derivatives) w.r.t. the elements of a symmetric tensor.

The result is a 6x6 matrix of the coefficients as used in gradient_transform(). The transformed gradients are obtained by multiplying this matrix with the vector of original gradients.

References gradient_transform_matrix().